# Inverse function: Pre-service teachers’ techniques and meanings

- 479 Downloads
- 2 Citations

## Abstract

Researchers have argued teachers and students are not developing connected meanings for function inverse, thus calling for a closer examination of teachers’ and students’ inverse function meanings. Responding to this call, we characterize 25 pre-service teachers’ inverse function meanings as inferred from our analysis of clinical interviews. After summarizing relevant research, we describe the methodology and theoretical framework we used to interpret the pre-service teachers’ activities. We then present data highlighting the techniques the pre-service teachers used when responding to tasks that involved analytical and graphical representations of functions and inverse functions in both decontextualized and contextualized situations and discuss our inferences of their meanings based on their activities. We conclude with implications for the teaching and learning of inverse function and areas for future research.

## Keywords

Function Inverse function Pre-service secondary teachers Meanings## Notes

### Acknowledgements

This material is based upon work supported by the NSF under Grant No. DRL-1350342. Any opinions, findings, conclusions, or recommendations expressed are those of the authors. We thank Neil Hatfield for directing us to a notation for inverse function and Nick Wasserman for his feedback on an earlier draft.

## References

- Bergeron, L., & Alcantara, A. (2015).
*IB mathematics comparability study: Curriculum & assessment comparison*. United Kingdom: UK NARIC. Retrieved from http://www.ibo.org/globalassets/publications/ib-research/dp/maths-comparison-summary-report.pdf. - Blyth, T. S. (2006).
*Lattices and ordered algebraic structures*. London, UK: Springer Science & Business Media.Google Scholar - Breidenback, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function.
*Educational Studies in Mathematics*,*23*(3), 147–185.Google Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics*(N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. & Trans.). Dordrecht: Kluwer Academic.Google Scholar - Brown, C., & Reynolds, B. (2007). Delineating four conceptions of function: A case of composition and inverse. In T. Lamberg & L. R. Wiest (Eds.),
*Proceedings of the 29th annual meeting of the north American chapter of the International Group for the Psychology of mathematics education*(pp. 190–193). Lake Tahoe, NV: University of Nevada, Reno.Google Scholar - Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. In A. H. Shoenfeld, J. J. Kaput, & E. Dubinsky (Eds.),
*Research in collegiate mathematics education, III: Issues in mathematics education*(Vol. 7, pp. 114–162). Providence: American Mathematical Society.Google Scholar - Carlson, M. P., Madison, B., & West, R. D. (2015). A study of students’ readiness to learn calculus.
*International Journal of Research in Undergraduate Mathematics Education*,*1*(2), 209–233.CrossRefGoogle Scholar - Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. A. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 547–589). Mahwah: Lawrence Erlbaum Associates.Google Scholar - Dugopolski, M. (2009).
*Fundamentals of precalculus*. Boston: Pearson Education.Google Scholar - Engelke, N., Oehrtman, M., & Carlson, M. P. (2005). Composition of functions: Precalculus students' understandings. In G. M. Lloyd, M. Wilson, J. L. M. Wilkins, & S. L. Behm (Eds.),
*Proceedings of the 27th annual meeting of the north American chapter of the International Group for the Psychology of mathematics education*(pp. 570–577). Roanoke, VA: Virginia Tech.Google Scholar - Even, R. (1992). The inverse function: Prospective teachers use of 'undoing'.
*International Journal of Mathematical Education in Science and Technology*,*23*(4), 557–562.Google Scholar - Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 517–545). Mahwah: Lawrence Erlbaum Associates, Inc.Google Scholar - Harel, G., & Sowder, L. (2005). Advanced mathematical-thinking at any age: Its nature and its development.
*Mathematical Thinking and Learning*,*7*(1), 27–50.CrossRefGoogle Scholar - Hunting, R. P. (1997). Clinical interview methods in mathematics education research and practice.
*The Journal of Mathematical Behavior*,*16*(2), 145-165.CrossRefGoogle Scholar - Larson, R., Hostetler, R. P., & Edwards, B. H. (2001).
*Precalculus with limits: A graphing approach*(3rd ed.). Boston: Houghton Mifflin Company.Google Scholar - Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching.
*Review of Educational Research*,*60*(1), 1–64.CrossRefGoogle Scholar - Lucus, C. A. (2005).
*Composition of functions and inverse function of a function: Main ideas as perceived by teachers and preservice teachers*(Unpublished doctoral dissertation). Canada: Simon Fraser University, BC.Google Scholar - Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment.
*Educational Studies in Mathematics*,*38*(1-3), 135–161.CrossRefGoogle Scholar - Merriam, S. B., & Tisdell, E. J. (2005).
*Qualitative research: A guide to design and implementation*. San Francisco: John Wiley & Sons.Google Scholar - Moore, K. C. (2014). Signals, symbols, and representational activity. In L. P. Steffe, K. C. Moore, L. L. Hatfield, & S. Belbase (Eds.),
*Epistemic algebraic students: Emerging models of students' algebraic knowing*(Vol. 4, pp. 211–235). Laramie: University of Wyoming.Google Scholar - Moore, K. C., Silverman, J., Paoletti, T., & LaForest, K. R. (2014). Breaking conventions to support quantitative reasoning.
*Mathematics Teacher Educator*,*2*(2), 141–157.CrossRefGoogle Scholar - Paz, T., & Leron, U. (2009). The slippery road from actions on objects to functions and variables.
*Journal for Research in Mathematics Education*,*40*(1), 18-39.Google Scholar - Phillips, N. (2015). Domain, co-domain and causation: A study of Britney’s conception of function. In T. Fukawa-Connelly, N. Infante, K. Keene, & M. Zandieh (Eds.),
*Proceedings of the 18th annual Conference on Research in Undergraduate Mathematics Education*(pp. 893–895). Pittsburgh: SIGMAA on RUME.Google Scholar - Piaget, J. (1968).
*Six psycological studies*(A. Tenzer, Trans.). New York: Random House.Google Scholar - Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. Kelly & R. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 267–306). Mahwah: Erlbaum.Google Scholar - Stewart, J., Redlin, L., & Watson, S. (2012).
*Precalculus: Mathematics for calculus, sixth edition*(6th ed.). Belmont: Brooks/Cole Cegage Learning.Google Scholar - Strauss, A. L., & Corbin, J. M. (1998).
*Basics of qualitative research: Techniques and procedures for developing grounded theory*(2nd ed.). Thousand Oaks: Sage Publications.Google Scholar - Sullivan, M., & Sullivan III, M. (2007).
*Precalculus: Concepts through functions, a unit circle approach to trigonometry*. Upper Saddle River: Pearson Education.Google Scholar - Swokowski, E. W., & Cole, J. A. (2012).
*Precalculus: Functions and graphs*(12 ed.). Belmont, CA: Cengage Learning.Google Scholar - Thompson, P., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.),
*The compendium for research in mathematics education*. Washington, DC: NCTM.Google Scholar - Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sépulveda (Eds.),
*Proceedings of the annual meeting of the International Group for the Psychology of mathematics education*(Vol. 1, pp. 31–49). Morélia: PME.Google Scholar - Thompson, P. W. (2013). Why use f(x) when all we really mean is y?
*OnCore, The Online Journal of the Arizona Association of Mathematics Teachers*, 18-26.Google Scholar - Thompson, P. W. (2016). Researching mathematical meanings for teaching. In L. D. English & D. Kirshner (Eds.),
*Handbook of international research in mathematics education*(pp. 435–461). New York: Taylor & Francis.Google Scholar - Thompson, P. W., Carlson, M. P., Byerley, C., & Hatfield, N. (2014). Schemes for thinking with magnitudes: A hypothesis about foundational reasoning abilities in algebra. In L. P. Steffe, K. C. Moore, L. L. Hatfield, & S. Belbase (Eds.),
*Epistemic algebraic students: Emerging models of students' algebraic knowing*(Vol. 4, pp. 1–24). Laramie: University of Wyoming.Google Scholar - Vidakovic, D. (1996). Learning the concept of inverse function.
*Journal of Computers in Mathematics and Science Teaching*,*15*(3), 295–318.Google Scholar - Vidakovic, D. (1997). Learning the concept of inverse function in a group versus individual environment. In E. Dubinsky, D. Mathews, & B. E. Reynolds (Eds.),
*Readings in cooperative learning for undergraduate mathematics*(Vol. 44, pp. 175–196). Washington, DC: The Mathematical Association of America.Google Scholar - von Glasersfeld, E. (1995).
*Radical constructivism: A way of knowing and learning*. Washington, DC: Falmer Press.Google Scholar - Zazkis, R., & Kontorovich, I. (2016). A curious case of superscript (− 1): Prospective secondary mathematics teachers explain.
*The Journal of Mathematical Behavior*,*43*, 98–110.CrossRefGoogle Scholar